From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited

C. Fairoos; Chiranjeeb Singha

REVIEW 4 major objections 5 minor 1 cited by

From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For extremal Reissner-Nordström and Kerr black holes, the Euclidean path integral cannot uniquely fix the entropy because neither conical regularity nor the Chern-Gauss-Bonnet theorem determines the periodicity of Euclidean time.

desk verdict Useful CGB extension for non-extremal RN and Kerr, but the extremal 'indeterminate entropy' claim is unsupported by the paper's own action calculation, which gives S=0. read the letter →

arxiv 2505.16309 v1 pith:DG46DVCV submitted 2025-05-22 gr-qc

C. Fairoos , Chiranjeeb Singha This is my paper

classification gr-qc MSC 83C5783C45 PACS 04.70.Dy

keywords blackholeentropyEuclideanpathintegralReissner-NordströmKerrextremalChern-Gauss-BonnettheoremHawking-GibbonsmethodEulercharacteristic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper revisits the Euclidean path integral derivation of black hole entropy for Reissner-Nordström and Kerr black holes and asks whether the periodicity of Euclidean time can be fixed by two standard tools: smoothness of the near-horizon geometry and the Chern-Gauss-Bonnet theorem. For non-extremal black holes both tools agree, producing the familiar inverse Hawking temperature and recovering the Bekenstein-Hawking area law. For extremal black holes, the near-horizon metric has no conical tip and the Euler characteristic of the relevant Euclidean section vanishes, so neither tool determines the period. The authors conclude that the entropy of extremal RN and Kerr black holes cannot be uniquely obtained from the Hawking-Gibbons path integral alone, which matters because extremal black holes are often treated as limits of non-extremal ones and their entropy remains contested.

What carries the argument

The load-bearing object is the Euclidean period $\beta$, the inverse temperature, fixed in the first method by demanding that the near-horizon metric has the flat polar form $dR^2+R^2d\theta^2$ with $\theta$ identified modulo $2\pi$, and in the second method by the Chern-Gauss-Bonnet theorem $\chi(M)=\frac{1}{32\pi^2}\int_M (R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})\sqrt{g}\,d^4x$, applied to the near-horizon Euclidean section. The theorem works when the section has $D^2\times S^2$ topology with $\chi=2$, converting the integral of curvature invariants into a value for $\beta$; it fails when the topology is $R\times S^1\times S^2$, where $\chi=0$ and the integral gives no information about $\beta$.

What would settle it

If a careful computation of the Euler characteristic of the complete asymptotically flat Euclidean extremal RN geometry, including boundary contributions at a large cutoff radius and then taking the cutoff to infinity, returned a nonzero value, then the CGB method could fix $\beta$ and the paper's conclusion would fail.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a dichotomy: when the Euclidean continuation of an RN or Kerr black hole has a $D^2\times S^2$ near-horizon topology, the condition of no conical singularity fixes $\beta = 2\pi r_+^2/\sqrt{M^2-Q^2}$ (RN) and $\beta = 2\pi(r_+^2+a^2)/(r_+-M)$ (Kerr), and the Chern-Gauss-Bonnet integral independently yields the same periods because $\chi=2$. The entropy then comes out as $S=\pi r_+^2$ or $S=\pi(r_+^2+a^2)$, one quarter of the horizon area. In the extremal limit, the near-horizon metric $ds^2 = (\rho^2/M^2)d\tau^2 + (M^2/\rho^2)d\rho^2$ cannot be put into the flat polar form $dR^2+R^2d\theta^2$, so no periodicity is forced by regularity; and the CGB integral vanishes because the Euclidean section has topology $R\times S^1\times S^2$. The paper therefore concludes that $\beta$, and hence the entropy, is not uniquely determined by either method for extremal RN and Kerr black holes.

Load-bearing premise

The argument assumes that the topology of the whole Euclidean black hole manifold is captured by its near-horizon slice, so boundary contributions to the Euler number at infinity can be dropped and the Euler characteristic is 2 for non-extremal cases and 0 for extremal cases.

Editorial extensions

If this is right

For non-extremal RN and Kerr black holes, both the near-horizon conical regularity condition and the Chern-Gauss-Bonnet integral give the same inverse temperature, and the entropy reduces to one quarter of the horizon area.
The Euclidean-time periodicity of a non-extremal RN or Kerr black hole is fixed by the Euler characteristic $\chi=2$ of the near-horizon $D^2\times S^2$ topology.
For extremal RN and Kerr black holes, the absence of a conical tip leaves the periodicity $\beta$ arbitrary, and the vanishing Euler characteristic ($\chi=0$ for $R\times S^1\times S^2$) removes the topological constraint.
Consequently the semiclassical Euclidean path integral alone does not determine a unique extremal black hole entropy; additional physical input is needed.
The consistency of the two methods in the non-extremal case supports the use of the CGB theorem as a topological derivation of temperature whenever the Euclidean section has the required compact-like topology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension, not attempted in the paper, would be to add a small nonzero temperature regulator and check whether the extremal entropy emerges from a limit that is independent of the regulator; the paper's reasoning suggests it would not within the same Euclidean method.
The same topology argument should apply to other extremal stationary black holes, such as Kerr-Newman or higher-dimensional Reissner-Nordström, whose near-horizon geometry is also $R\times S^1\times S^2$ (or an analogue); computing their CGB integral would be a direct test.
If the ambiguity is structural rather than technical, then microscopic derivations of extremal entropy cannot be interpreted as evaluating the bulk Euclidean action on a smooth extremal saddle; they must be supplying information from outside the semiclassical geometry.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new thing here is the explicit Chern-Gauss-Bonnet treatment of RN and Kerr, following Hughes's recent Schwarzschild computation. For non-extremal cases, the periods from conical regularity and from CGB agree, and the entropy reduces to A/4. That part is sound and a real extension of the topological route.

The extremal section has a genuine problem. The paper concludes that because the Euclidean time period is not fixed by either method, the entropy cannot be uniquely determined. But that does not follow from the action calculation shown. Using the paper's own Eq. 12 on the extremal RN metric gives I = βM, not the βM/2 in Eq. 30; the factor is off. With either expression, the action is linear in β with a fixed coefficient, so applying S = β∂I/∂β − I gives S = 0, not an undetermined value. The CGB result χ=0 only says the topology does not select a period; it says nothing about the entropy. To defend the 'no unique entropy' conclusion, the authors would need to show that the action itself has no well-defined β-dependence, or to discuss the limit-ordering between extremal and non-extremal. As written, the paper's own equations give S=0 for the extremal case.

Smaller issues: the step from Eq. 15 to Eq. 16 is not shown; the reader cannot see how the thermodynamic identity is applied. The Kerr CGB integral is quoted without intermediate steps, which makes independent checking hard. The compactness caveat for CGB is handled too quickly, though the falloff argument for RN is plausible.

The stress-test's specific concern about the Maxwell boundary term is not the crux. The paper's action without the A-boundary term corresponds to a canonical ensemble and does produce the correct area law; the real flaw is the extremal inference from undetermined β to undetermined S.

This is a readable, honest paper that cites the relevant literature. The non-extremal material is a good review, and the CGB extension is worth having out there. But the extremal claim as stated is not supported by the displayed equations. Yes, send it to peer review—referee time is justified because the topological route is of interest and the extremal entropy issue is mainstream. The referee should ask the authors to correct Eq. 30, show the thermodynamic algebra, and either retract or properly argue the indeterminacy claim.

Referee Report

4 major / 5 minor

Summary. The paper revisits the Euclidean (Gibbons–Hawking) path-integral derivation of black hole entropy for Reissner–Nordström and Kerr black holes, and then examines the extremal limit. For non-extremal black holes, the Euclidean time periodicity β is fixed by two methods: the requirement that the near-horizon geometry has no conical singularity, and an application of the Chern–Gauss–Bonnet (CGB) theorem through the Euler characteristic of the Euclidean section. The authors find that both methods give the usual Hawking temperature and, via the thermodynamic relation S = β ∂I/∂β − I, the Bekenstein–Hawking area law S = A/4. For extremal black holes, the near-horizon geometry has no conical tip, so β is not fixed by regularity; the CGB integral also gives a vanishing Euler characteristic, so the topological method likewise fails to constrain β. The paper concludes that the entropy of extremal RN and Kerr black holes cannot be uniquely determined within the Euclidean path-integral framework.

Significance. If the derivations were fully correct, the paper would provide a clear and useful comparison of the conical-regularity and topological routes to black hole temperature for the two canonical charged and rotating families, and it would sharpen the well-known but often under-explained statement that the Euclidean method does not uniquely fix extremal black hole entropy. The non-extremal results reproduce the standard area law, and the conical-regularity calculations are explicit, machine-checkable in principle, and free of fitted parameters. The CGB computations for Schwarzschild and non-extremal RN are also explicit and consistent. However, the central extremal claim—that the action method leaves the entropy indeterminate—is not fully established by the displayed equations, because the boundary-term/ensemble convention is not specified and one displayed action value is algebraically inconsistent. The extremal Kerr CGB statement is asserted without calculation. These are fixable issues, and they do not undermine the non-extremal results, which are independently supported by the conical-regularity method.

major comments (4)

[III, Eqs. (15)–(16)] The step from Eq. (15) to Eq. (16) is not self-contained and requires the β-dependence of r_+. As displayed, I_E in Eq. (15) is proportional to β with M and Q held fixed, so a naive application of S = β∂I/∂β − I would give zero, not β/2√(M²−Q²). One must differentiate using Eq. (18), where r_+ depends on β. In addition, Eq. (15) is the on-shell action in a fixed-charge convention without a Maxwell boundary term; adding the standard boundary term −(1/4π)∫ n_μ F^{μν} A_ν changes the action to β/2√(M²−Q²). The authors should state the ensemble, show the differentiation explicitly, and verify that the final entropy is ensemble-independent.
[IV, Eq. (30)] The extremal RN action I_E = βM/2 is inconsistent with Eq. (15) evaluated at Q = M, r_+ = M, which gives I_E = βM. If a different boundary-term convention is being used, that convention should be stated and applied consistently. The qualitative conclusion that S is undetermined does survive for an action linear in β with arbitrary β, but with the grand-canonical boundary term one obtains I_E = 0 and hence S = 0. The non-uniqueness claim is therefore boundary-condition dependent and should be presented as such.
[IV, extremal Kerr paragraph] The claim that the CGB integral for extremal Kerr 'vanishes identically' is asserted without any supporting computation. In contrast to the extremal RN case, where Eq. (32) gives the explicit integral, no analogous expression is provided for Kerr. Since the failure of the CGB method to constrain β in the Kerr case is one of the paper's two central extremal results, the integral should be written out and evaluated.
[II, Eq. (9); III, Eq. (19); IV, Eq. (32)] The application of the CGB theorem to noncompact, asymptotically flat Euclidean sections is not fully justified. The paper acknowledges that the theorem requires a compact manifold, but it never defines the compact truncation or shows that the boundary contributions to the Euler characteristic vanish; instead it assigns χ = 2 or χ = 0 from the assumed near-horizon topology. For the non-extremal cases this is corroborative because the conical-regularity method independently fixes β, but for the extremal argument the statement that 'the CGB theorem fails to constrain β because χ = 0' requires that the bulk integral is the only contribution to χ. Please either supply the boundary-term analysis or reformulate the claim as conditional on the assumed near-horizon topology.

minor comments (5)

[II, paragraph after Eq. (18)] The symbol 'χ(⇕ E)' appears to be a typo for χ(M_E); please correct it.
[Throughout] There are several typographical issues, including 'Wicks' rotated metric instead of 'Wick-rotated', and the encoding of 'Reissner-Nordstr¨om' in the abstract; a careful proofreading pass is needed.
[References] Reference [21] is incomplete: it lists no journal, volume, year, or arXiv identifier, so the reader cannot verify the origin of the CGB method used here.
[III, Kerr action] The transition from the boundary integral for Kerr to the final result I_E = βM/2 is not shown; in particular, the fate of the logarithmic term in the integrated expression at r → ∞ should be displayed explicitly.
[IV, Eq. (31)] The statement that the near-horizon extremal metric 'can be identified with any period β' would benefit from a sentence explaining that ρ = 0 is at infinite proper distance, so the shrinking τ-circle does not create a conical singularity for any β.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the periodicity inputs are independent of the entropy output; the cited CGB compactness assumption comes from an external paper, not from the authors' own prior claims.

full rationale

I walked the derivation chain. In Sections II and III, the Euclidean time periodicity β is fixed by two independent conditions: (i) the absence of a conical singularity in the near-horizon geometry, Eqs. (18) and (27), and (ii) the topological Euler characteristic of the Euclidean section, χ = 2 for D2 × S2, in the CGB integrals, Eq. (19) and the Kerr analogue. Neither input presupposes the Bekenstein-Hawking area law or the numerical value S = A/4; the χ = 2 input is a topological statement about a disk times a sphere, not an entropy value. The entropy is then obtained from the standard relation S = β∂I/∂β − I after β is fixed, and the result S = πr_+^2 follows from evaluating that expression, not from fitting a parameter to the desired answer. For the extremal cases, β is left undetermined because the near-horizon metric has no conical-singularity constraint for arbitrary period and the CGB integral gives χ = 0; again, this conclusion comes from the geometry and topology of the extremal section, not from the desired entropy value. The paper borrows the near-horizon CGB application from Hughes et al. [21], but that is an external citation rather than a self-citation and does not smuggle in the target entropy. No fitted parameter is later renamed as a prediction, and no displayed equation uses the claimed output as an input. The algebraic inconsistency between Eq. (15) and Eq. (16) noted by the skeptic is a correctness or typographical concern, not circularity, because the displayed entropy value is never used to fix β anywhere in the derivation. Accordingly, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or new entities. Its central conclusion rests on the semiclassical Euclidean path integral, the extension of CGB to noncompact spaces, and the assumed topology of the Euclidean sections. These are the main assumptions a reader needs to audit.

assumptions (4)

domain assumption The Euclidean path integral is dominated by classical saddle points, so the partition function is e^{-I_E}.
Introduced in Section II around Eq. (3); this is the semiclassical approximation underlying all entropy derivations in the paper.
ad hoc to paper The Chern-Gauss-Bonnet theorem can be applied to noncompact Euclidean sections by integrating bulk curvature only, with boundary terms vanishing.
The paper notes compactness is required (Section II) but follows Hughes [21] in restricting to the near-horizon regime and dropping boundary contributions; this is the weakest unproved step.
domain assumption Non-extremal Euclidean RN and Kerr sections have topology D2 x S2 with Euler characteristic 2; extremal sections have topology R x S1 x S2 with Euler characteristic 0.
Used in Section II (Schwarzschild), Section III (RN, Kerr), and Section IV (extremal) to convert chi into beta or to conclude that chi = 0 gives no constraint.
standard math The thermodynamic relation S = beta dI/dbeta - I holds, with M treated as an implicit function of beta through the smoothness condition.
This is the standard Gibbs-Hawking relation, Eq. (2), used throughout; in the text the derivative step is often skipped.

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Pith. "Pith review of From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited." pith.science (2026). https://pith.science/paper/DG46DVCV

@misc{pith2026250516309,
  author       = {Pith},
  title        = {Pith review of: From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DG46DVCV}},
  note         = {Machine review of arXiv:2505.16309}
}

read the original abstract

In this paper, we derive the entropy of Reissner-Nordstr\"om (RN) and Kerr black holes using the Hawking-Gibbons path integral method. We determine the periodicity of the Euclidean time coordinate using two approaches: first, by analyzing the near-horizon geometry, and second, by applying the Chern-Gauss-Bonnet (CGB) theorem. For non-extremal cases, both these methods yield a consistent and unique periodicity, which in turn leads to a well-defined expression for the entropy. In contrast, the extremal case exhibits a crucial difference. The absence of a conical structure in the near-horizon geometry implies that the periodicity of the Euclidean time is no longer uniquely fixed within the Hawking-Gibbons framework. The CGB theorem also fails to constrain the periodicity, as the corresponding Euler characteristic vanishes. As a result, the entropy cannot be uniquely determined using either method.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Black hole thermodynamics and topology
gr-qc 2025-05 reject novelty 4.0 of 10

The author derives S_RN = 4πM^2 for Reissner-Nordström black holes by adding the inverse temperatures of both horizons, contradicting the standard area law.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Bekenstein, J. D. Black Holes and Entropy. Phys. Rev. D 1973, 7, 2333–2346

work page 1973

[2] [2]

Hawking, S. W. Particle Creation by Black Holes. Commun. Math. Phys. 1975, 43, 199–220, [Erratum: Com- mun.Math.Phys. 46, 206 (1976)]

work page 1976

[3] [3]

W.; Hawking, S

Gibbons, G. W.; Hawking, S. W. Action integrals and parti tion functions in quantum gravity. Phys. Rev. D 1977, 15, 2752–2756

work page 1977

[4] [4]

Entropy and action of di laton black holes

Kallosh, R.; Ort ´ ın, T.; Peet, A. Entropy and action of di laton black holes. Phys. Rev. D 1993, 47, 5400–5407

work page 1993

[5] [5]

Can extremal black ho les have nonzero entropy? Mod

Das, S.; Dasgupta, A.; Ramadevi, P. Can extremal black ho les have nonzero entropy? Mod. Phys. Lett. A 1997, 12, 3067–3080

work page 1997

[6] [6]

W.; Horowitz, G

Hawking, S. W.; Horowitz, G. T.; Ross, S. F. Entropy, Area , and black hole pairs. Phys. Rev. D 1995, 51, 4302–4314

work page 1995

[7] [7]

W.; Horowitz, G

Hawking, S. W.; Horowitz, G. T. The Gravitational Hamilt onian, action, entropy and surface terms. Class. Quant. Grav. 1996, 13, 1487–1498

work page 1996

[8] [8]

Microscopic origin of the Beken stein-Hawking entropy

Strominger, A.; Vafa, C. Microscopic origin of the Beken stein-Hawking entropy. Phys. Lett. B 1996, 379, 99–104

work page 1996